The Magicians

by Marcus Chown

  For almost a month, Dirac told nobody about his discovery; he broke his silence only on the eve of leaving Cambridge for his parents’ home in Bristol for the Christmas vacation, when he bumped into Charles Galton Darwin, grandson of the biologist and a leading theoretical physicist. Darwin was deeply impressed by what Dirac told him, and on Boxing Day wrote to the Danish quantum physicist Niels Bohr: ‘[Dirac] has now got a completely new system of equations which does the spin right in all cases and seems to be “the thing”.’

  Dirac submitted a paper to the Royal Society on 1 January 1928, and it appeared in print a month later.16 ‘The Quantum Theory of the Electron’ caused a sensation. According to the American physicist John Van Vleck, Dirac’s explanation of an electron’s spin was comparable to ‘a magician’s extractions of rabbits from a silk hat’.

  The successes of Dirac’s equation, however, came at a cost. At the outset, he had attempted to exclude the possibility of negative-energy electrons, but he had failed miserably. His beautiful equation contained not one but two sets of ‘two-by-two’ matrices – one representing positive-energy electrons and the other negative-energy electrons.§

  In ‘classical’, or pre-quantum, physics, it was not unusual for a theory to throw up such nonsensical ‘solutions’, but physicists simply dismissed them by saying that nature chooses not to implement them. In quantum theory, however, no such option is available; according to American Nobel Prize-winning physicist Murray Gell-Mann, ‘Everything that is not forbidden is compulsory.’ In other words, a particle such as an electron has a non-zero ‘probability’ of making a ‘transition’ from any state to any other state – and that included Dirac’s negative-energy states.

  Dirac had discovered that Einstein’s special theory of relativity could be satisfied by electrons only if they had spin, which was an enormous triumph, but he had also discovered it could be satisfied only if electrons were permitted to have both positive and negative energies, which was disastrous.

  Physicists were amazed by the beauty of Dirac’s equation and stunned by its power to predict things about the real world, but many were unsettled by its prediction of negative-energy electrons. To Werner Heisenberg, this was evidence that the equation was sick and quite possibly wrong. ‘The saddest chapter of modern physics is and remains the Dirac theory,’ he wrote despairingly to Wolfgang Pauli, who agreed. In Pauli’s opinion, the sickness of Dirac’s equation was incurable, and the agreement of its predictions with experiments was little more than a fluke.

  Dirac himself did not share the misgivings of other physicists at the troubling negative-energy feature of his equation. Although his forte was the most abstract fundamental physics, he had trained as an electrical engineer and was at his core a pragmatist. If something worked – and his equation worked in predicting to unprecedented levels of precision much that had been observed in experiments – then he was sure it must contain a large amount of truth. If it failed in some respects, it might simply need some tweaking, so all he had to do was find a way to do that.

  A major reason for the despair of Heisenberg was that the negative energy ‘solutions’ of the Dirac equation threatened the very stability of matter. In the everyday world, objects tend to reduce their ‘potential energy’ – that is, energy with the potential to, in physicists’ jargon, do ‘work’. For instance, given the chance, a ball at the top of a hill will race to the bottom, converting its potential energy into energy of motion. At the top of the hill, it is said to have ‘high gravitational potential energy’, and at the bottom ‘low gravitational potential energy’.

  The problem with Dirac’s equation was that, if the negative-energy ‘states’ were available to electrons, there was nothing to stop them minimising their potential energy by dropping into those states. It was as inevitable as a ball rolling to the foot of a hill: matter would be unstable. Dirac’s equation spelled catastrophe for the world.

  Things did not look good, but in the autumn of 1928, Dirac came up with a radical idea for avoiding the disaster. Arguably, it was one of the most ridiculous ideas in the history of science.

  Matter is stable, he pointed out, so all the electrons in the universe have not, by definition, dropped into the negative energy states. The obvious explanation was that his equation was wrong and that the negative-energy states did not exist, but the equation had scored so many spectacular successes that he did not want to abandon it. He therefore proposed an alternative explanation for why the electrons in matter have not dropped into the negative-energy states. There is no room for them. Why? Because those states are already filled to the brim with negative-energy electrons.¶

  The fact that the idea appeared nuts was not necessarily grounds to dismiss it. The key question was: Did it contradict reality? Surely we would notice if we are living in the midst of a vast sea of negative-energy electrons, but Dirac reasoned that we would not. Do we in normal circumstances notice the air around us? Do fish notice the water through which they swim?

  By postulating a vast sea of negative-energy electrons to fix the difficulty with the stability of matter, Dirac was able to sweep a major problem of his equation under the carpet, but in doing so he created another headache. A vast sea of negative-energy electrons would, not surprisingly, have consequences. Occasionally, for instance, a negative-energy electron might be struck by a photon; if ejected from the sea with sufficient energy, it would become a normal positive-energy electron.

  The sudden appearance in the world of an electron like a rabbit plucked out of a hat was a startling enough notion, but when Dirac followed his reasoning through to its logical conclusion, he realised something else: the ejected electron would leave behind a gaping hole in the sea of negative-energy electrons. He knew of experiments in which the inner electron in an atom was ejected by a high-energy ‘X-ray’ photon and left behind a similar absence; in these cases, this ‘hole’ behaved exactly like a positively charged electron. Dirac proposed that the hole left behind when an electron is ejected from the negative-energy sea behaves exactly like a positively charged particle. In other words, a high-energy photon would create not one particle but two: an electron plus a positively charged mirror image of an electron, in a process that would become known as ‘pair production’.

  Dirac was not brave enough to propose the existence of an new subatomic particle with a mass equivalent to an electron but an opposite electric charge on the basis of a mathematical formula he had conjured out of thin air, so he chose a more cautious option. At the time, the only known positively charged subatomic particle was the proton, so Dirac proposed that the positively charged mirror image of the electron was a proton. The fact that such a particle is about two thousand times the mass of an electron, marring the neat symmetry of pair production, was a detail to be sorted out later. A new fundamental particle would have been surplus to requirements and would have been strongly resisted by physicists. It was a battle Dirac did not wish to fight – so he dodged it.

  According to Dirac’s friend Peter Kapitza, he never seriously believed that his positively charged particle was a proton. He proposed it simply so he would not have to face other physicists mocking him with the question, ‘Where is your antielectron, Professor Dirac?’

  In actual fact, the proton was never a serious contender for the mirror image of the electron conjured into existence in pair production. The American physicist Robert Oppenheimer, who would one day lead the Manhattan Project to build an atomic bomb, pointed out that if a high-energy photon could create an electron and a proton, then the reverse process would also be possible, with a proton and an electron annihilating each other. This would cause matter to be dangerously unstable. Atoms would survive only as long as their protons did not run into stray electrons. At every instant, they would be prone to disappearing in a flash of gamma rays.

  It was largely Oppenheimer’s argument that emboldened Dirac to go public with what he already knew in his bones. In May 1931, he submitted another paper to the Royal Society. It wa
s on a different subject entirely – a speculation on why electric charge comes in discrete chunks, or ‘quanta’; however, in the paper Dirac predicted ‘the existence of a new kind of particle, unknown to experimental physics, having the same mass as an electron but opposite charge’.17 He called it an ‘antielectron’ and wrote, ‘We should not expect to find any of them in nature, on account of their rapid rate of recombination with electrons, but if they could be produced experimentally in high vacuum they would be quite stable and amenable to observation.’

  During a lecture at Princeton University in late October 1931, Dirac went further. ‘Antielectrons are not to be considered as the mathematical fiction,’ he said. ‘It should be possible to detect them by experimental means.’18

  What Dirac imagined were two high-energy photons colliding and conjuring an electron and an antielectron into existence. He was not optimistic about the imminent detection of such a process, since photons of the extremely high energy required were unlikely to be available to experimenters for the foreseeable future. Dirac must have been aware that cosmic rays possessed extremely high energies – typically thousands of times higher than those of the particles spat out by the nuclei of radioactive atoms – and that they might create antielectrons when they slammed into particles in the atmosphere, yet he appeared to give them little thought. This was possibly because the experimenters he knew at Cambridge considered them insufficiently interesting to study and thought Millikan was wasting his time.

  Dirac did not only predict a positively charged partner of the electron; in his Royal Society paper of May 1931, he pointed out that just as a relativistic description of the electron implied the existence of an antielectron, a relativistic description of the proton implied the existence of an antiproton. Nature must have duplicated all its fundamental particles, and there existed a mirror world of positive electrons and negative protons – a universe of ‘antimatter’. ‘My equation’, Dirac later confessed, ‘was smarter than I was.’19

  Dirac had well and truly stuck his neck out in writing down an equation, motivated by nothing more than a desire to make quantum theory and special relativity mathematically consistent, which predicted much of what physicists observed in the world, including the existence of quantum spin. But remarkably, it also predicted that the stuff of the world that had hitherto seemed immutable – the fundamental particles of matter – could be created and destroyed at will. And if that was not shocking enough, in order for such processes to occur, there must exist a mirror universe of antimatter.

  Rarely in the history of science has a single equation predicted so much novelty. ‘There is something fascinating about science,’ Mark Twain observed. ‘One gets such wholesale returns of conjecture out of such a trifling investment of fact.’20 Of no piece of science has that been more true than the Dirac equation.

  Postulating the existence of a subatomic particle that nobody had ever seen and for which there had never been any need was controversial to say the least, but the proof of the pudding would be in the eating. For Dirac, the big question was: Did antielectrons really exist?

  Pasadena, California, Autumn 1932

  Carl Anderson had found no other tracks showing a positively charged particle with the mass of an electron. It was such a worry that he considered asking the journal Science to withdraw his paper. Had he done so, however, it would have been too late; the printing presses were already rolling.

  On 1 September 1932, the paper appeared, and the reaction from other physicists was either indifference or outright disbelief. Ed McMillan, a good friend from Anderson’s undergraduate days at Caltech, waved his copy of Science under Anderson’s nose. ‘What sort of nonsense is this?’ he asked. Millikan, who had become convinced that there was something wrong with the cosmic ray experiment for it to yield such an incomprehensible result, was not supportive either. Anderson, his confidence undermined, wondered whether he had been a total fool and whether, at the age of twenty-seven, he had inadvertently sabotaged his scientific career.

  Perhaps if Anderson had realised what he had found, it might have made a difference, but he had no idea that the particle he had detected had been predicted at a desk in Cambridge by Paul Dirac. Weirdly, he had recently been attending evening lectures given by Oppenheimer, who spent several months of each year at Caltech, and they had dealt with Dirac’s hole theory at length. Anderson failed to make the connection between a Dirac hole and the peculiar particle he had discovered in his Guggenheim cloud chamber, but Oppenheimer’s own blindness was arguably even more bizarre. Despite knowing about Dirac’s prediction of a positively charged electron and Anderson’s discovery of a positively charged electron, he unaccountably failed to put the two things together.

  One of Anderson’s colleagues did make the connection. Rudolph Langer, a mathematician, knew about Dirac’s theory of the antielectron and had seen Anderson’s photograph of the track of a lightweight, positively charged particle. Shortly after reading Anderson’s paper in Science, he submitted a short response to the journal in which he categorically claimed that the particle Anderson had detected was Dirac’s antielectron. Unfortunately, Langer was neither well known nor respected in physics circles and his paper was ignored.

  It was even worse six thousand miles away in Cambridge, where nobody seemed to be aware of Anderson’s experiment nor of Langer’s paper in Science. It would take an independent experiment to wake up the physicists at the Cavendish Laboratory.

  Patrick Blackett had belatedly got into cosmic ray research after Millikan had lectured at Cambridge the previous year and shown some intriguing cloud chamber photographs taken at Caltech that were, of course, Anderson’s. Blackett persuaded the Cavendish Laboratory director, Ernest Rutherford – the greatest experimental physicist of the age and discoverer of the atomic nucleus – to let him get into cosmic ray research. He teamed up with the Italian physicist Giuseppe Occhialini, and the pair hit on the clever idea of observing the debris from cosmic rays by using Geiger-Müller tubes in conjunction with a cloud chamber.

  A Geiger-Müller tube, or Geiger counter, consists of a gas-filled glass tube. When a particle of radiation passes through it, it knocks electrons from molecules of the gas, which are amplified by a high voltage into a measurable electric current. By putting one Geiger counter above their cloud chamber and one below it, and triggering the chamber only if both Geiger counters registered a current, Blackett and Occhialini ensured that every photograph they took contained particle tracks. Whereas Anderson’s experiment had been a lottery, with the odds of detecting positrons stacked against him, this experiment was a dead cert, and the particles were photographed in large numbers.

  Dirac was always hazy about how he heard about the discovery of antielectrons, but he probably heard about them from Blackett. The pictures he had obtained with Occhialini of positrons in cosmic ray showers were so sensational that they were featured on the front pages of newspapers. By mid-December 1932, there could no longer be any doubt, and Dirac confirmed that the pictures of pair production obtained at the Cavendish Laboratory were consistent with his theory. His days of panic attacks were over, and no fact, either experimental or theoretical, was going to wreck his beautiful equation. His triumph was to predict for the first time in scientific history the existence of a new fundamental particle. And he had done it using a theory he had pulled out of thin air, with pretty much no motivation from any experiment.

  Blackett and Occhialini had undoubtedly provided the best evidence of the existence of the positron. In fact, Blackett had observed the positron’s effects even earlier than Anderson, though he made the mistake of dismissing them as unimportant. Despite his own pivotal contribution, Blackett was always scrupulously careful to give credit to Anderson for being the first to announce the existence of the positron.21

  In 1936, Anderson was rewarded with the Nobel Prize in Physics, sharing it with Hess, the discoverer of cosmic rays. By this time, Dirac had also been honoured, having shared the 1933 Nobel Prize with
Schrödinger.

  Anderson would be part of a Nobel Prize-winning dynasty of three generations of experimental physicists. Not only did his supervisor, Robert Millikan, win the prize but so too did his student, Donald Glaser, who carried off the 1950 prize for the invention of the ‘bubble chamber’, which revealed the tracks of subatomic particles in a similar way to the cloud chamber.

  *

  With hindsight, pair production and the existence of antimatter should have come as no surprise to anyone. It turns out that something like it is essential to unite the quantum and relativistic descriptions of the electron, or of any subatomic particle.

  One of the cornerstones of physics is that energy can neither be created nor destroyed, but only transformed from one form into another. In a world ruled by special relativity, where mass itself is a form of energy, this ‘law of conservation of energy’ has an unavoidable consequence. The energy of motion of photons can be transformed into the mass-energy of subatomic particles – creating matter – and the mass-energy of subatomic particles can be converted into the energy of motion of photons – destroying matter.

  But quantum theory applies a crucial restriction on the processes of creation and destruction: electric charge, like energy, cannot be created or destroyed. The ‘law of conservation of electric charge’ means that, in the creation of matter, a photon, which has no electric charge, cannot change into a subatomic particle which has an electric charge. However, a photon can change into two identical particles which carry opposite charges, so that their net charge is zero. Similarly, in the destruction of matter, a charged particle cannot change into a photon. This requires two identical particles with opposite charge. Thus we are led to the idea that the creation of matter must involve a photon spawning a particle and an antiparticle – pair production – and the destruction of matter must involve a particle and antiparticle spawning a photon – annihilation (yet another restriction, known as the ‘law of conservation of momentum’, dictates that the annihilation of matter and antimatter must result in two identical, oppositely directed photons).